Susie Boyt has adored Judy Garland since her first visit to the cinema, where she heard Dorothy singing 'Somewhere Over the Rainbow'. As a lonely and sensitive child, every word Garland sang spoke directly to her. She took the singer to heart, wanting to love her and somehow become her. Now an adult, 'the imperative intimacy I feel with Judy Garland is similar to that which I feel for my mother and my children'. This is partly playful, partly deadly serious. Boyt truly believes in the moral dimension to her relationship with Judy Garland. It represents a belief in life and a belief in a better self, which can drag you out of the hellish bits of the everyday.

My Judy Garland Life follows the trajectory of hero-worship, telling the story of one girl's attempt to overcome loneliness. But it has higher aims. The lessons Boyt has learnt from Garland are not small ones. 'All feelings, however painful, are to be prized. Glamour is a moral stance. Loss, its memory and its anticipation, lies at the heart of human experience.'

This desire to create something profound out of something seemingly superficial makes for an extremely strange but rather wonderful undertaking. The book defies definition. It has elements of biography, autobiography, self-help and fan letter; although it's firmly non-fiction, there are parts that read like a novel. Above all, it is a bold experiment that sets out to map the boundaries of celebrity obsession, and somewhere along the way discovers what it means to be human. It is a defence of Judy Garland, as well as a defence of Boyt herself.

It cannot help but be all the more fascinating because of Boyt's family connections. Freud was her great-grandfather and she grew up surrounded by adults who talked about psychoanalysis. Her father is painter Lucian Freud. Her parents separated before she was born and she rarely saw her father as a child. From the earliest age, she almost encouraged herself to feel others' pain. So it hardly seems surprising that she identified so strongly with Judy Garland.

Despite the pathos in both stories, there is plenty of vaudeville and slapstick here, playing up to Boyt's fantasy that somehow she, too, could lead a Garland life, given half a chance. She learnt to dance as a girl ('Between 1976 and 1983, I attended almost 2,000 dancing classes') and was bitterly disappointed when her mother refused to let her attend the Italia Conti school because she was 'too intelligent'. She can be very funny, listing all the moments in her life when she has received back-handed compliments about her lack of talent as a singer and a dancer. She tells a friend she would have loved to be in musical theatre. This doesn't count as a regret, comes the reply, 'because to qualify as a regret something has to have been within the realms of possibility'. Another friend pours cold water on her fantasy of appearing in an amateur musical in a prison by warning that the standard would be too high.

Boyt paints herself as the best kind of fan: admiring, adoring, but remaining at a respectful distance. But throughout the book, hovering in the background, tacitly acknowledged, is the spectre of the stalker. She knows she could easily overstep the mark; she would have loved to watch Garland sleep, she admits, as Diana Ross once asked to do. And when she finally meets Liza Minnelli, Garland's daughter, she has to restrain herself from collecting her cigarette butts and the remains of her club sandwich.

What is most charming about this book is how brazen and irreverent it is. Boyt does not care if we don't like Judy Garland ('Come on, who doesn't love a show tune!' you can hear her cry). She doesn't mind if the book doesn't fit into a particular genre. She just wants to write her story. This devil-may-care confidence fits in well with the Garland myth: the book, like life, has a 'show must go on' quality, poised delicately between desperation and joy. It may well also be the first book ever to feature a mention of the website 'They Died on the Toilet' within a page of a reference to James Joyce. Boyt has created a universe where Derrida rubs shoulders with Sinatra and her great-grandfather Sigmund hangs out with Liza Minnelli. It is a fabulous place to be and, like an old Hollywood feelgood movie, at first I didn't want it to stop.

By the close of My Judy Garland Life, though, I wasn't quite sure what, apart from beautiful, heartstopping writing, had pushed me to read to the end. Judy Garland died five months after Boyt was born so there is no real plot, no potential for encounter, aside perhaps from her quest to meet Liza Minnelli, Mickey Rooney and various other people connected to Garland.

As there is no mystery to solve, what you are left with is gaps. Boyt seems apprehensive about delving too deeply into what her obsession really says about her relationship with her father. Judy Garland only exists as an absence in her life, just as her father was in her childhood. She only hints at the difficulties she has endured in this regard, including a troubling episode when her father promised to do a painting for her and then didn't. But perhaps this is part of the beauty of the book. Boyt gives away more about her real reasons for allowing Judy Garland so far into her life by not saying very much about them at all.

The most revealing moment comes when she tries to ask Liza Minnelli directly about her mother. Instead, Boyt suddenly finds herself confessing that she is in awe of her own father and that other people's excessive interest in him makes her very uncomfortable. 'But don't you understand where they're coming from?' Minnelli replies.